1998 OIM Problems/Problem 6

Problem

Let $\lambda$ be the positive root of the equation $t^2 - 1998t - 1 = 0$ . The sequence $x_0, x_1, x_2, \cdots , x_n, \cdots$ is defined by:

$\begin{cases} x_0=1 \\ x_{n+1}=\left\lfloor \lambda x_n \right\rfloor\ , & \text{for }n=0,1,2,\cdots\end{cases}$

Find the remainder of the division of $x_{1998}$ by $1998$ .

Note: The brackets indicate an integer part, that is,

$\left\lfloor x \right\rfloor$

is the only integer  such that .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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